SOLUTION: an enclosure is to be constructed having part of its boundary along an existing straight wall. the other part of the boundary is to be fenced in the shape of an arc of a circle. if

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Question 1173312: an enclosure is to be constructed having part of its boundary along an existing straight wall. the other part of the boundary is to be fenced in the shape of an arc of a circle. if 100m of fencing is available, what is the area of the largest possible enclosure? into what fraction of the circle is the fence bent?
Found 3 solutions by ankor@dixie-net.com, greenestamps, ikleyn:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
an enclosure is to be constructed having part of its boundary along an existing straight wall.
the other part of the boundary is to be fenced in the shape of an arc of a circle.
if 100m of fencing is available, what is the area of the largest possible enclosure? into what fraction of the circle is the fence bent?
:
I think the greatest area would be if the fence was shaped in a semi-circle with the wall.
Find the radius of a semicircle with an arc length of 100
pi%2Ar = 100
r = 100%2Fpi
find the area of a circle with this radius
A = pi%2A%28100%2Fpi%29%5E2
A = pi%2A%28100%5E2%2Fpi%5E2%29
for a semicircle (cancel pi)
A = 1%2F2*100%5E2%2Fpi
A = 1591.55 sq/meters

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The solution from the other tutor is of no use.

He "thinks" that is the answer; and then his calculations are not correct.

MY thought is that the problem can't be answered unless the radius of the circle is known.

If the radius of the circle is equal to 100%2F%282pi%29, or about 15.9, then the 100m of fencing will enclose a complete circle.

If the radius of the circle is 50m, then the 100m of fencing will just reach across the circle, and the fence will be a straight line along the existing wall, making the area of the enclosure zero.

A radius somewhere between those two extremes of 15.9m and 50m will produce an enclosure of maximum area, with part of the boundary along the existing wall.

Perhaps the intent of the problem is in fact to determine the radius that produces the maximum area of the enclosure; but that seems to be a problem that can't be solved by any method I know.

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

If you reflect the shape,  which you are looking for,  about the wall as a mirror,
you reduce the given problem to the following problem 

    find a shape / (a figure) in the plane, which at given perimeter of 200 m
    has maximum possible area.


It is  WELL  KNOWN  classic problem on finding maximum area at given perimeter,  and its solution is known  VERY  WELL: 

    The solution is a circle with the given circumference  (perimeter) of  200 m.


Keeping it in mind,  every student of the  6th - 7th grade can complete the solution of the given problem
on his  (or her)  own without any difficulties.


The answer is     a semi-circle of the radius of   200%2F%282pi%29 = 100%2Fpi = 100%2F3.14159 = 31.831 meters

with the area of   %281%2F2%29%2Api%2A%28100%2Fpi%29%5E2 = %281%2F2%29%2A%2810000%2Fpi%29 = 5000%2F3.14159 = 1591.551 square meters.


Happy learning  (!)


For your education,  see this Wikipedia article

https://en.wikipedia.org/wiki/Area     ,  the section  "Circle area".